Glossary

5.1 Principles of seismic ray theory

3 min read•august 9, 2024

Seismic ray theory explains how waves travel through Earth's layers. It uses principles like and to describe wave paths and travel times. These concepts help geologists understand Earth's structure and locate earthquakes.

Ray theory simplifies complex wave behavior into straight-line paths. It's useful for modeling seismic waves in different media types, from simple isotropic layers to complex anisotropic structures. This approach is key to interpreting seismic data and imaging Earth's interior.

Principles of Ray Theory

Fundamental Principles of Wave Propagation

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  • Fermat's principle states waves travel along paths that minimize between two points
  • Applies to various types of waves (light, sound, seismic)
  • Explains phenomena like and in different media
  • Mathematically expressed as δ∫ABn(r)ds=0\delta \int_{A}^{B} n(r) ds = 0, where n(r) represents refractive index
  • Snell's law describes relationship between angles of incidence and refraction for waves passing through boundary between two media
  • Expressed as sin⁡θ1sin⁡θ2=v1v2\frac{\sin \theta_1}{\sin \theta_2} = \frac{v_1}{v_2}, where θ represents angles and v represents velocities

Wave Propagation Mechanisms

  • explains wave propagation through construction of wavefronts
  • States every point on a acts as a source of secondary wavelets
  • New wavefront envelope forms from these secondary wavelets after a time t
  • Accounts for wave phenomena like diffraction and interference
  • Wavefront represents surface of constant phase in a wave
  • Perpendicular to direction of wave propagation
  • Types include plane wavefronts (parallel lines) and spherical wavefronts (concentric circles)
  • Ray paths always perpendicular to wavefronts in

Ray Parameterization

Ray Parameter and Its Significance

  • (p) represents constant quantity along in layered media
  • Defined as p=sin⁡θvp = \frac{\sin \theta}{v}, where θ represents angle with vertical and v represents
  • Remains constant for a given ray as it travels through different layers
  • Useful in calculating travel times and determining ray paths
  • Helps in solving inverse problems in seismology (determining Earth's structure from seismic data)

Mathematical Foundations of Ray Theory

  • describes high-frequency approximation of wave equation
  • Fundamental equation in ray theory, expressed as (∇T)2=1v2(\nabla T)^2 = \frac{1}{v^2}
  • T represents travel time and v represents velocity
  • Solves for travel times of waves in heterogeneous media
  • Ray tracing involves calculating paths of seismic waves through Earth's interior
  • Uses principles of ray theory to determine travel times and wave paths
  • Methods include shooting method (trial and error) and bending method (minimizing travel time)
  • Applications in , earthquake location, and studying Earth's internal structure

Media Types

Isotropic Media Characteristics

  • Isotropic media exhibit uniform physical properties in all directions
  • Wave velocity remains constant regardless of propagation direction
  • Simplifies wave equations and ray tracing calculations
  • Examples include homogeneous fluids and some crystalline solids (cubic crystals)
  • Snell's law and ray parameter remain valid in layered isotropic media
  • Wave polarization remains constant during propagation through isotropic media

Anisotropic Media Complexities

  • display direction-dependent physical properties
  • Wave velocity varies with propagation direction
  • Complicates wave equations and ray tracing procedures
  • Causes phenomena like shear wave splitting and azimuthal velocity variations
  • Examples include layered sedimentary rocks and minerals with preferred orientations
  • Requires tensor representation of physical properties (elasticity, conductivity)
  • Leads to more complex wave behavior, including quasi-P and quasi-S waves
  • Important in studying Earth's mantle structure and crustal deformation

Key Terms to Review (20)

Acoustic Impedance: Acoustic impedance is a measure of how much resistance a medium offers to the passage of sound waves through it. It is defined as the product of the medium's density and the speed of sound in that medium. This concept is crucial for understanding how seismic waves interact with different geological layers, affecting the reflection and refraction at interfaces.
Anisotropic media: Anisotropic media are materials that exhibit different properties depending on the direction of measurement. This means that seismic waves travel at varying speeds and exhibit different behaviors in different directions within the material. Understanding anisotropic media is crucial in seismic ray theory, as it affects how seismic waves propagate through the Earth's subsurface, influencing the interpretation of seismic data and the analysis of subsurface structures.
Critical Angle: The critical angle is the specific angle of incidence above which a seismic wave cannot pass through a boundary and is instead reflected back into the medium. This concept is vital in understanding how seismic waves interact with different layers of the Earth, influencing the paths they take and how they can be detected at the surface.
Density: Density is defined as the mass per unit volume of a material, typically expressed in grams per cubic centimeter (g/cm³) or kilograms per cubic meter (kg/m³). This property is crucial for understanding how seismic waves travel through different materials within the Earth, as density affects wave speed and behavior.
Eikonal Equation: The eikonal equation is a fundamental equation in wave propagation theory that describes the relationship between the wavefronts of a propagating wave and the path taken by a ray of that wave. It mathematically characterizes how waves travel through different media, illustrating the principle that the travel time is related to the geometric characteristics of the wavefronts. This equation is essential in seismic ray theory, providing a framework for understanding how seismic waves propagate through complex geological structures.
Fermat's Principle: Fermat's Principle states that the path taken by a ray of light between two points is the path that requires the least time. This principle is foundational in understanding how waves, including seismic waves, propagate through different media, which is crucial for seismic ray theory.
Huygens' Principle: Huygens' Principle states that every point on a wavefront can be considered as a source of secondary wavelets, which spread out in all directions at the same speed as the original wave. This principle helps to explain how waves propagate through different media and relates directly to concepts like velocity models, ray paths, and the behavior of waves during reflection and refraction.
Incidence angle: The incidence angle is defined as the angle between an incoming seismic wave and a surface or boundary that it encounters, typically measured from the normal (perpendicular) to that surface. Understanding this angle is crucial in seismic ray theory because it influences how waves reflect, refract, and transmit through different geological layers, affecting the interpretation of subsurface structures.
Isotropic media: Isotropic media are materials that have the same physical properties in all directions. This uniformity means that seismic waves travel at consistent velocities regardless of the wave's propagation direction. In the context of seismic ray theory, isotropic media simplify the analysis of wave propagation, allowing for clearer predictions about how seismic waves will behave as they travel through different layers of the Earth's crust.
Medium interface: A medium interface refers to the boundary or surface where two different materials or mediums meet, impacting the propagation of seismic waves. When seismic waves travel from one medium to another, their speed, direction, and intensity can change due to differences in properties such as density and elastic modulus at this interface. Understanding medium interfaces is crucial in analyzing how seismic waves are transmitted through the Earth's layers and interpreting seismic data.
Ray parameter: The ray parameter is a measure that describes the angle of incidence of seismic rays traveling through the Earth, expressed in terms of the horizontal distance traveled per unit of vertical distance. It provides insights into how seismic waves propagate through different layers of the Earth's crust, allowing for the construction of ray paths and the analysis of travel time curves. Understanding the ray parameter is essential for interpreting seismic data and evaluating the subsurface structure.
Ray path: A ray path is the trajectory that seismic waves follow as they travel through the Earth's interior and along its surface. This concept is fundamental in understanding how seismic waves propagate and how they can be used to infer properties about the Earth's structure. The analysis of ray paths helps seismologists determine the location and magnitude of seismic events, as well as provides insights into subsurface geological formations.
Reflection: In seismology, reflection refers to the bouncing back of seismic waves when they encounter a boundary between different types of geological materials. This process is crucial for understanding the internal structure of the Earth, as it helps identify different layers and their properties by analyzing how seismic waves behave at these boundaries.
Refraction: Refraction is the bending of seismic waves as they pass through different layers of the Earth's interior, caused by variations in wave speed due to changes in material properties. This phenomenon is crucial for understanding how seismic waves travel and interact with different geological structures, which aids in identifying seismic phases, analyzing travel time curves, and interpreting seismograms.
Seismic imaging: Seismic imaging is a technique used to visualize the internal structure of the Earth by analyzing the reflected and refracted seismic waves generated by earthquakes or artificial sources. This method provides crucial insights into subsurface geology, allowing scientists to map geological formations and understand the Earth's composition, which is essential for various applications in Earth science.
Snell's Law: Snell's Law describes how seismic waves change direction when they pass through different layers of material with varying properties, specifically concerning their velocities. This fundamental principle is crucial for understanding how waves refract and reflect as they encounter boundaries within the Earth's subsurface, influencing methods of data interpretation in seismology.
Tomography: Tomography is a imaging technique that uses seismic waves to create detailed cross-sectional images of the Earth's internal structure. By analyzing the travel times and paths of seismic waves generated by earthquakes or artificial sources, tomography helps to visualize subsurface features and variations in material properties. This method plays a crucial role in understanding geological formations, from the shallow crust to deep mantle and core structures.
Travel Time: Travel time refers to the duration it takes for seismic waves to propagate from their source, such as an earthquake, to a recording station. Understanding travel time is crucial for interpreting seismic data, as it helps determine the distance to the seismic event and informs the analysis of wave behavior in different geological settings.
Velocity: In seismology, velocity refers to the speed at which seismic waves travel through different materials in the Earth. This concept is crucial for understanding how waves propagate and interact with geological structures, influencing methods used to interpret subsurface conditions and locate seismic events.
Wavefront: A wavefront is an imaginary surface representing points of a wave that oscillate in unison, moving through a medium. This concept is crucial in understanding how seismic waves propagate, interact with different geological structures, and change direction as they encounter boundaries between materials with varying properties.
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